A restricted second-order logic for non-deterministic poly-logarithmic time. (12th May 2020)

• Record Type: Journal Article
• Authors: Ferrarotti, Flavio; GonzÁles, SenÉn; Schewe, Klaus-Dieter; Turull-Torres, JosÉ MarÍa
• Is Part Of: Logic journal of the IGPL. Volume 28:Number 3(2020)
• Page Start: 389
• Page End: 412
• Publication Date: 2020-05-12
• Subjects: descriptive complexity -- poly-logarithmic time -- second-order logic -- finite models; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1093/jigpal/jzz078
• Languages: English
• ISSNs: 1367-0751

Abstract

Abstract: We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin's style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin's theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail.